As discussed in Section REF , the magnetic Dirac phase enters
in the definition of the \(2^+\) -summable even \((\) pre-\()\) Fredholm module
\((\rho ,{H}_4,F_{B,\varepsilon })\)  (Theorem REF ). It is worth mentioning
that the latter fact allows to associate to \(F_{B,\varepsilon }\)  a unique \(K\!K\) -homology
class \([F_B]\in K\!K({C}_B,\)  (see [1]}),
although this aspect will not be investigated further in this work. Another
important remark is that with the help of the
differential (REF ) one can build a quantized calculus
for the magnetic algebra \({C}_B\)  in the spirit of [2]}. This aspect will be investigated in full detail in an upcoming work.
In the present work, we provide only a preliminary, but related result, known
in [3]} with the name of first Connes'
formula. For that, we need to recall that the pre-\(C^*\) -algebra \({S}_B\)  admits
two spatial derivations defined by the commutators with the position operators,
i.e.,
\(\nabla _j(A) = -\,\mathrm {i}\,[x_j,A] ,\qquad A\in {S}_B ,\qquad j=1,2 .\) 
